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How Do the Roots of a Polynomial Depend on the Coefficients?

The literature is full of methods for finding the roots of polynomials;
however, few if none ask how well the roots can be determined, even with a
perfect root-finding method. Since the coefficients of the polynomial are most
likely stored as floating-point numbers, the floating-point resolution leads to
an absolute bound on the accuracy of the roots. Uncertainty of the inputs to
the root finding lead to uncertainty in the outputs.

Assume that we have already found a root z of a polynomial
P(x), so that

Multiple Roots

Note that this expression is invalid when the derivative of the polynomial at z is zero. This is exactly the case
where the polynomial has multiple roots at z. There are two distinct cases for multiple roots. The first is when the
roots are fundamentally independent and just happen to coincide. One example of this is from optical raytracing, where
a ray can just graze a surface. For this sort of polynomial, the above result is correct; the polynomial is unstable with
respect to the coefficients, and the slightest change can alter the root from real to complex.

The alternative is when the roots are structurally multiple. An example of this is when the polynomial was created as
the square of another polynomial. In this case, the dependence of the coefficients can be found by looking at the
derivative. Define a new polynomial Q(z):

Q(z)=∑PkP0kzk−1Q(z) = SUM { {P sub k} over {P sub 0} } k z sup {k-1}

(8)

If the root in question is double, then the sensitivity can be found by using Q(z) in place of P(z) in equation (7). For higher multiplicities, just repeat the derivative process on Q(z).

Numerical Analysis

Now define δ as the relative distance between two adjacent floating point numbers. In other words, if a real number is rounded to the nearest floating point number N, N would collect up all the numbers in the range [ (1-δ/2)N,(1+δ/2)N]. The maximum error in a polynomial term is then

Emax,k=δ2|Pk|E sub {max,k} = %delta over 2 abs { P sub k }

(9)

and the rms error, assuming that the errors are uniformly distributed, is